Graph Theory: Euler Graphs and Digraphs Pallab Dasgupta, Professor, Dept. of Computer Sc. and Engineering, IIT Kharagpur pallab@cse.iitkgp.ernet.in

Euler Circuit We use the term circuit as another name for closed trail. A circuit containing every edge of G is an Eulerian circuit. A graph whose edges comprise a single closed trail is Eulerian.

Properties Non-trivial maximal trails in even graphs are closed. A finite graph G is Eulerian if and only if all its vertex degrees are even and all its edges belong to a single component. For a connected nontrivial graph with 2k odd vertices, the minimum number of pairwise edge-disjoint trails covering the edges is max{k, 1}.

Fleury’s Algorithm Input: A graph G with one non-trivial component and at most two odd vertices. Initialization: Start at a vertex that has odd degree unless G is even, in which case start at any vertex. Iteration: From the current vertex, traverse any remaining edge whose deletion from the graph does not leave a graph with two non-trivial components. Stop when all edges have been traversed.

Euler Trails in Directed Graphs Input: A digraph G that is an orientation of a connected graph and has d+(u) = d(u) for all u V(G). Step1: Choose a vertex v V(G). Let G be the digraph obtained from G by reversing direction on each edge. Search G to construct T consisting of paths from v to all other vertices. Step2: Let T be the reversal of T. T contains a u,v-path in G for each u  V(G). Specify an arbitrary ordering of the edges that leave each vertex u, except that for uv, the edge leaving u in T must come last. Step3: Construct an Eulerian circuit from v as follows. Whenever u is the current vertex, exit along the next unused edge in the ordering specified for edges leaving u.

The Chinese Postman Problem Suppose a mail carrier traverses all edges in a road network, starting and ending at the same vertex. The edges have non-negative weights representing distance or time. We seek a closed walk of minimum total length that uses all the edges.